Explanation Given: The vector-valued functions r ( t ) = 〈 2 + cos t , 3 + sin 2 t , t 〉 ; t 0 = π 2 . Definition used: Let r ( t ) = 〈 f ( t ) , g ( t ) , h ( t ) 〉 be the vector-valued function. Its derivative can be given by r ′ ( t ) = 〈 f ′ ( t ) , g ′ ( t ) , h ′ ( t ) 〉 where f , g and h are differentiable functions. Derivative rule: (1) Constant rule: d d t ( c ) = 0 . (2) d d t ( cos a t ) = − a sin a t [ a is constant ] . (3) d d t ( sin a t ) = a cos a t [ a is constant ] . Calculation: The derivative of the vector-valued function r ( t ) = 〈 2 + cos t , 3 + sin 2 t , t 〉 ; t 0 = π 2 is computed as follows, r ′ ( t ) = d d t ( r ( t ) ) = d d t 〈 2 + cos t , 3 + sin 2 t , t 〉 = 〈 d d t ( 2 + cos t ) , d d t ( 3 + sin 3 t ) , d d t ( t ) 〉 Apply the constant rule and derivative rules (2) and (3), r ′ ( t ) = 〈 d d t ( 2 + cos t ) , d d t ( 3 + sin 2 t ) , d d t ( t ) 〉 = 〈 ( 0 − sin t ) , ( 0 + 2 cos 2 t ) , 1 〉 = 〈 − sin t , 2 cos 2 t , 1 〉 Substitute t = π 2 in r ′ (

3rd Edition

ISBN: 9780134856971

Author: Briggs

Publisher: PEARSON

See similar textbooks

Concept explainers

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magn…

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.

Question

Chapter 14.2, Problem 81E

To determine

To find: An equation of the line parallel to the curve at t=t0 .

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.2, Problem 80E

Chapter 14.2, Problem 82E

Students have asked these similar questions

Exer. 27-32: (a) Use f' to prove that f has an inverse function. (b) Find the slope of the tangent line at the point P on the graph of f. 27 f(x) = x² + 3x³ + 2x-1; P(5, 1)

Find the maximum rate of change of f at the given point and the direction in which it occurs.

Fill in the blanks to complete each of the following theoremstatements: The derivative of a vector function r(t) is given by r'(t) =limh→0__________________. The Chain Rule for Vector-Valued Functions: Let t = f (τ) bea differentiable real-valued function of τ, and let r(t) bea differentiable vector function with either two or threecomponents such that f (τ ) is in the domain of r for everyvalue of τ on some interval I. Then dr/dτ =_______________ . Let C be the graph of a twice-differentiable vector functionr(t) defined on an interval I and with unit tangent vector T(t). Then the curvature κ of C at a point on the curve is given by κ = ||__________|| / ||__________|| and κ = ||_____x______||/||_________||3 Let y = f (x) be a twice-differentiable function. Then thecurvature of the graph of f is given by κ =| | / ( )3/2 Let C be the graph of a vector function r(t) = ⟨x(t), y(t)⟩ inthe xy-plane, where x(t) and y(t) are twice-differentiablefunctions of t such that x'(t)…

Chapter 14 Solutions

CALCULUS: EARLY TRANSCENDENTALS (LCPO)

Ch. 14.1 - Restrict the domain o f the vector function in...Ch. 14.1 - Explain why the curve in Example 5 lies on the...Ch. 14.1 - How many independent variables does the function...Ch. 14.1 - How many dependent scalar variables does the...Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - How do you evaluate limtar(t), where r(t) = f(t),...Ch. 14.1 - How do you determine whether r(t) = f(t) i + g(t)...Ch. 14.1 - Find a function r(t) for the line passing through...Ch. 14.1 - Find a function r(t) whose graph is a circle of...

Additional Math Textbook Solutions

Find more solutions based on key concepts

29-36. Total and Annual Returns. Compute the total and annual returns on the following investments. 29. Five ye...

Using and Understanding Mathematics: A Quantitative Reasoning Approach (6th Edition)

1. combination of numbers, variables, and operation symbols is called an algebraic______.

Algebra and Trigonometry (6th Edition)

Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval est...

Elementary Statistics (13th Edition)

2. Source of Data In conducting a statistical study, why is it important to consider the source of the data?

Elementary Statistics

4. You construct a 95% confidence interval for a population mean using a random sample. The confidence interval...

Elementary Statistics: Picturing the World (7th Edition)

Three cards are randomly selected, without replacement, from an ordinary deck of 52 playing cards. Compute the ...

A First Course in Probability (10th Edition)

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

An equation of the line parallel to the curve at t = t 0 .

Concept explainers

To find: An equation of the line parallel to the curve at t=t0 .

Want to see the full answer?

Chapter 14 Solutions

Additional Math Textbook Solutions